L(s) = 1 | + (−0.950 + 0.548i)2-s + (−0.328 + 0.189i)3-s + (−0.397 + 0.689i)4-s + (0.208 − 0.360i)6-s − 1.89i·7-s − 3.06i·8-s + (−1.42 + 2.47i)9-s + 0.134·11-s − 0.301i·12-s + (−3.04 − 1.75i)13-s + (1.03 + 1.79i)14-s + (0.887 + 1.53i)16-s + (1.43 − 0.830i)17-s − 3.13i·18-s + (−2.10 − 3.81i)19-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.388i)2-s + (−0.189 + 0.109i)3-s + (−0.198 + 0.344i)4-s + (0.0849 − 0.147i)6-s − 0.715i·7-s − 1.08i·8-s + (−0.476 + 0.824i)9-s + 0.0405·11-s − 0.0871i·12-s + (−0.843 − 0.487i)13-s + (0.277 + 0.480i)14-s + (0.221 + 0.384i)16-s + (0.348 − 0.201i)17-s − 0.738i·18-s + (−0.483 − 0.875i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.323881 - 0.256428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.323881 - 0.256428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.10 + 3.81i)T \) |
good | 2 | \( 1 + (0.950 - 0.548i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.328 - 0.189i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 1.89iT - 7T^{2} \) |
| 11 | \( 1 - 0.134T + 11T^{2} \) |
| 13 | \( 1 + (3.04 + 1.75i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.43 + 0.830i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.65 + 2.68i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 1.69iT - 37T^{2} \) |
| 41 | \( 1 + (5.31 + 9.20i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.36 + 4.25i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (9.62 + 5.55i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.229 - 0.132i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.44 + 5.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.55 + 1.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.664 + 1.15i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.49 - 3.17i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.733 + 1.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44iT - 83T^{2} \) |
| 89 | \( 1 + (-4.86 + 8.43i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.1 - 8.73i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41132740547362323669991773436, −10.07905016965892593885257502176, −8.862446936790064984391918334860, −8.053598457440517350217376788597, −7.39620641342059961584364757425, −6.40920121812608843732073436733, −5.03536542818867235233648573996, −4.08669882197482173357171720911, −2.62233553175053946952169566869, −0.32675783309284023612900439178,
1.54590032027215215792444074912, 2.91391673133943453371331947510, 4.54028728371632479635282683741, 5.70351477065145822507756275078, 6.39331864313757986224006154045, 7.88372303389859251829077904446, 8.676589685479094881050052536708, 9.537767164854965813043512224913, 10.08070691893961092739694087906, 11.19959374268840667411647026326